Limits
Lessons in this chapter
About Limits
Limits opens AP Calculus AB by building the foundational idea calculus is built on: what a function approaches, even at a point it never quite reaches.
Introduction to Limits and One-sided and Infinite Limits define what a limit means, including limits approached from just one side or that grow without bound, before Numerical Estimation and Oscillating Behavior covers estimating a limit from a table of values, including functions that oscillate too fast to have a limit at all.
Limits by Direct Substitution and Limit Laws and Evaluating Limits Algebraically cover the most direct techniques for calculating a limit, while Trig Limits and Strategies for Finding Limits and Limit of Piecewise and Composite Functions extend those techniques to trigonometric, piecewise, and composite functions.
Squeeze Theorem introduces a technique for finding a limit by trapping a function between two others with the same limit, and Limits at Infinity closes the chapter by covering what a function approaches as its input grows infinitely large.
Nine lessons, each a practice worksheet worked in the page, run from Introduction to Limits through Evaluating Limits Algebraically to Limits at Infinity. Every problem is scored as you answer it, so you find out whether you can evaluate a limit by hand instead of watching a solver hand you the value. Free to try, with no login needed for your first 3.
Frequently asked questions
Introduction to Limits describes what a function approaches rather than what it reaches. One-sided and Infinite Limits and Numerical Estimation and Oscillating Behavior cover approaching from a single side and the cases where nothing settles, Limits by Direct Substitution and Limit Laws and Evaluating Limits Algebraically give the methods for when substitution works and for when it does not, and Trig Limits and Strategies for Finding Limits and Limit of Piecewise and Composite Functions handle the harder forms. Squeeze Theorem and Limits at Infinity close with trapping a limit between two others and with end behavior. This is Unit 1 of the AP Calculus AB course framework, Limits and Continuity, and it goes past anything the Common Core covers in high school.
A limit describes the value a function approaches as its input gets closer and closer to a specific point, whether or not the function actually reaches that value there.
A one-sided limit only considers the input approaching from one direction (left or right), and an infinite limit describes a function growing without bound as the input approaches a point.
Plug in input values increasingly close to the target point and watch the output values, but a function that oscillates too rapidly near that point may never settle on a single approaching value at all.
Direct substitution plugs the target value straight into the function; when that works, it’s usually the fastest way to evaluate a limit.
Factoring, rationalizing, and simplifying the expression first often removes the issue (like a 0/0 form) that direct substitution ran into.